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import sympy
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from sympy import symbols, limit, diff, Eq, solve
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from convert_formula import convert_formula
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def compute_limit(f, vars, points, direction=None):
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"""
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计算多元函数的重极限或累次极限
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Args:
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f (sympy.Expr): 待计算极限的多元函数表达式
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vars (list[sympy.Symbol]): 函数中的变量列表（如[x, y]）
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points (list[float]): 极限点的坐标列表（与vars顺序对应，如[1, 2]）
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direction (str, optional): 累次极限方向，可选'x_first'（先x后y）或'y_first'（先y后x），默认None表示计算重极限
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Returns:
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sympy.Expr: 计算得到的极限值
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"""
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if direction == 'x_first':
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lim_x = limit(f, vars[0], points[0])
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return limit(lim_x, vars[1], points[1])
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elif direction == 'y_first':
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lim_y = limit(f, vars[1], points[1])
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return limit(lim_y, vars[0], points[0])
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else:
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return limit(f, *zip(vars, points))
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def check_continuity(f, vars, points):
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"""
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检查多元函数在指定点的连续性（极限值是否等于函数值）
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Args:
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f (sympy.Expr): 待检查的多元函数表达式
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vars (list[sympy.Symbol]): 函数中的变量列表
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points (list[float]): 待检查点的坐标列表（与vars顺序对应）
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Returns:
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bool: 连续返回True，不连续返回False
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"""
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f_value = f.subs(dict(zip(vars, points)))
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lim = compute_limit(f, vars, points)
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return Eq(lim, f_value)
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def partial_derivative(f, vars_list):
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"""
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计算多元函数对指定变量列表的高阶偏导数（按顺序求导）
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Args:
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f (sympy.Expr): 待求导的多元函数表达式
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vars_list (list[sympy.Symbol]): 求导的变量列表（按求导顺序排列，如[x, y]表示先对x再对y求导）
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Returns:
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sympy.Expr: 计算得到的高阶偏导数表达式
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"""
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return diff(f, *vars_list)
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def total_differential(f, vars):
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"""
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计算多元函数的全微分表达式
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Args:
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f (sympy.Expr): 待计算全微分的多元函数
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vars (list[sympy.Symbol]): 函数中的所有变量列表
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Returns:
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sympy.Expr: 全微分表达式（如df = f_x*dx + f_y*dy）
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"""
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diffs = [diff(f, var) for var in vars]
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return sum(d * sympy.Derivative(var) for d, var in zip(diffs, vars))
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def directional_derivative(f, vars, points, direction_vec):
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"""
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计算多元函数在指定点沿给定方向的方向导数
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Args:
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f (sympy.Expr): 待计算的多元函数
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vars (list[sympy.Symbol]): 函数中的变量列表
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points (list[float]): 计算点的坐标列表（与vars顺序对应）
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direction_vec (list[float]): 方向向量（如[1, 1]表示45度方向）
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Returns:
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float: 方向导数的数值结果
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"""
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grad = gradient(f, vars)
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grad_val = [g.subs(dict(zip(vars, points))) for g in grad]
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dir_unit = [v / sympy.sqrt(sum(d**2 for d in direction_vec)) for v in direction_vec]
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return sum(g * d for g, d in zip(grad_val, dir_unit))
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def gradient(f, vars):
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"""
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计算多元函数的梯度向量
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Args:
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f (sympy.Expr): 待计算的多元函数
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vars (list[sympy.Symbol]): 函数中的变量列表（如[x, y]）
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Returns:
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list[sympy.Expr]: 梯度向量（各分量为对应变量的偏导数）
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"""
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return [diff(f, var) for var in vars]
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def find_extrema(f, vars):
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"""
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寻找多元函数的无条件极值点并判断极值类型
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Args:
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f (sympy.Expr): 待分析的多元函数
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vars (list[sympy.Symbol]): 函数中的变量列表
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Returns:
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list[tuple[dict, str]]: 极值点信息列表，每个元素为(极值点字典, 极值类型)
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"""
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critical_points = solve([diff(f, var) for var in vars], vars, dict=True)
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hessian = sympy.hessian(f, vars)
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extrema = []
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for point in critical_points:
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hess_val = hessian.subs(point)
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eigvals = hess_val.eigenvals()
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if all(v > 0 for v in eigvals):
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extrema.append((point, '极小值'))
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elif all(v < 0 for v in eigvals):
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extrema.append((point, '极大值'))
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else:
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extrema.append((point, '鞍点'))
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return extrema
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def constrained_extrema(f, constraints, vars):
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"""
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使用拉格朗日乘数法求解带约束条件的极值问题
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Args:
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f (sympy.Expr): 目标函数（待求极值的函数）
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constraints (list[sympy.Expr]): 约束条件列表（如[Eq(x**2+y**2, 1)]）
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vars (list[sympy.Symbol]): 函数中的变量列表
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Returns:
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list[dict]: 满足约束条件的极值点解（包含拉格朗日乘子）
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"""
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lambdas = symbols('lambda:' + str(len(constraints)))
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lagrangian = f - sum(l * c for l, c in zip(lambdas, constraints))
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equations = [diff(lagrangian, var) for var in vars] + constraints
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variables = vars + list(lambdas)
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solutions = solve(equations, variables, dict=True)
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return solutions
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def UI():
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vars_dict = {'x': symbols('x'), 'y': symbols('y'), 'z': symbols('z')}
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while True:
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print("\n多元函数导数分析系统")
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print("1. 计算重极限/累次极限")
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print("2. 检查连续性")
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print("3. 计算偏导数")
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print("4. 计算全微分")
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print("5. 计算方向导数")
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print("6. 计算梯度")
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print("7. 寻找无条件极值")
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print("8. 求解条件极值")
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print("0. 退出")
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choice = input("请选择功能(0-8): ")
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if choice == '0':
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print("系统已退出")
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break
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try:
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expr = convert_formula(input("请输入多元函数表达式(如x**2+y**2): "))
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f = sympy.sympify(expr)
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vars_list = list(f.free_symbols)
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if choice in ['1','2','5','7']:
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points = [float(p) for p in eval(input("请输入极限点/求值点坐标(用,分隔): "))]
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if choice == '1':
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direction = input("是否计算累次极限？(x_first/y_first/直接回车为重极限): ")
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res = compute_limit(f, vars_list, points, direction)
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print(f"计算结果: {res}")
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elif choice == '2':
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res = check_continuity(f, vars_list, points)
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print(f"连续性判断: {res}")
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elif choice == '3':
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vars_input = input("请输入求偏导的变量名（用,分隔，如x,y表示先对x再对y求导）: ")
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vars_list = [vars_dict.get(var.strip(), symbols(var.strip())) for var in vars_input.split(',')]
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res = partial_derivative(f, vars_list)
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print(f"偏导数结果: {res}")
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elif choice == '4':
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res = total_differential(f, vars_list)
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print(f"全微分结果: {res}")
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elif choice == '5':
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dir_vec = [float(v) for v in eval(input("请输入方向向量(用,分隔): "))]
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res = directional_derivative(f, vars_list, points, dir_vec)
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print(f"方向导数结果: {res}")
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elif choice == '6':
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res = gradient(f, vars_list)
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print(f"梯度结果: {res}")
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elif choice == '7':
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res = find_extrema(f, vars_list)
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print(f"无条件极值点: {res}")
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elif choice == '8':
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constraints_expr = convert_formula(input("请输入约束条件表达式(用,分隔): "))
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constraints = [sympy.sympify(c) for c in constraints_expr.split(',')]
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res = constrained_extrema(f, constraints, vars_list)
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print(f"条件极值解: {res}")
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except Exception as e:
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print(f"输入错误: {str(e)}")
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if __name__ == '__main__':
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UI() |
